Converting from Rectangular Coordinates to Polar - Concept

Converting from rectangular coordinates to polar coordinates. And that can be kind of tricky because remember that the polar coordinates for a point are not unique. So let's make a rule here that we're going to get r to be greater than or equal to 0 and theta between 0 and 2 pi. This will allow us to get a unique set of polar coordinates for a point and there will be a lot less confusion. Your teacher may actually have a requirement like that as well.
So all of these are rectangular coordinates. I want to convert them to polar. So let's start with this one. So x is root 3 and y is 3. Now the first thing I can do is use this formula here to find the r value, right? The distance of the point from the pole. So r squared is root 3 squared plus 3 squared. That's 3+9 12. So r=2 root 3. Remember you should the positive value for r so there's no plus or minus here. We want the positive value. And then we can use these formulas to find theta. So you have x=r cosine theta and that means cosine theta equals x over r. Now x is root 3, r is 2 root 3. And so that's one half. And I also use this formula, right? y=r sine theta. So sine theta is y over r. The y value is 3, the r value is 2 root 3 and I can rationalize this denominator by multiplying by root 3 over root 3 and I get 3 root 3 over 2 times 3. The threes cancel. Root 3 over 2. So the question is what angle has a cosine of one half and a sine of root 3 over 2?
Well, that's pi over 3. Theta's pi over 3. And remeber we want theta to be between 0 and 2 pi. I mean there is another angle of 7 pi over 3 that works but it's not in the interval that we want. So this is the theta that we want and the polar coordinates are going to be, remember r comes first, so 2 root 3 theta. Pi over 3.
Okay, let's try another example. The point -5 5. First, r squared equals x squared plus y squared. So -5 squared plus 5 squared. That's 25+25 or 50. And that means that r equals 5 root 2. Again, we picked up the positive value. And we're going to use the fact that cosine theta is x over r, -5 over 5 root 2 and that's -1 over root 2 which is the same as -2 over root 2 over 2. And sine theta is y over r. So it's 5 over 5 root 2, 1 over root 2 which is root 2 over 2. Now what angle has a cosine that's negative root 2 over 2 and a sine that's positive root 2 over 2? You could draw a little unit circle if you like. We're over here, right? A negative a negative cosine value and a positive sine value. This is going to be a reference angle of 45 degrees. The angle's 135 which in radiance is 3 pi over 4. Now that means my point is going to be r first 5 root 2 theta, 3 pi over 4.
And finally let's do 0 -10. Actually this one's pretty easy. If you if you graph this point, you can kind of see that well, we can't really use an angle of negative pi over 2 because it's not in the interval that we want. But you can use an angle of 3 pi over 2. So theta would be 3 pi over 2. And remember that r is just a distance from the origin, which is 10. So this would be 10 3 pi over 2. So it's much easier if you're converting a point that is on a coordinate axis in the polar coordinates.